Discrete Mathematics with Applications

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Susanna Epp's DISCRETE MATHEMATICS, THIRD EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses.

The Logic of Compund Statements

1

(74)

Logical Form and Logical Equivalence

1

(16)

Statements

Compound Statements

Truth Values

Evaluating the Truth of More General Compound Statements

Logical Equivalence

Tautologies and Contradictions

Summary of Logical Equivalences

Conditional Statements

17

(12)

Logical Equivalences Involving →

Representation of If-Then As Or

The Negation of a Conditional Statement

The Contrapositive of a Conditional Statement

The Converse and Inverse of a Conditional Statement

Only If and the Biconditional

Necessary and Sufficient Conditions

Remarks

Valid and Invalid Arguments

29

(14)

Modus Ponens and Modus Tollens

Additional Valid Argument Forms: Rules of Inference

Fallacies

Contradictions and Valid Arguments

Summary of Rules of Inference

Application: Digital Logic Circuits

43

(14)

Black Boxes and Gates

The Input/Output for a Circuit

The Boolean Expression Corresponding to a Circuit

The Circuit Corresponding to a Boolean Expression

Finding a Circuit That Corresponds to a Given Input/Output Table

Simplifying Combinational Circuits

NAND and NOR Gates

Application: Number Systems and Circuits for Addition

57

(18)

Binary Representation of Numbers

Binary Addition and Subtraction

Circuits for Computer Addition

Two's Complements and the Computer Representation of Negative Integers

8-Bit Representation of a Number

Computer Addition with Negative Integers

Hexadecimal Notation

The Logic of Quantified Statements

75

(50)

Introduction to Predicates and Quantified Statements I

75

(13)

The Universal Quantifier: A

The Existential Quantifier: E

Formal Versus Informal Language

Universal Conditional Statements

Equivalent Forms of the Universal and Existential Statements

Implicit Quantification

Tarski's World

Introduction to Predicates and Quantified Statements II

88

(9)

Negations of Quantified Statements

Negations of Universal Conditional Statements

The Relation among A, E, V, and V

Vacuous Truth of Universal Statements

Variants of Universal Conditional Statements

Necessary and Sufficient Conditions, Only If

Statements Containing Multiple Quantifiers

97

(14)

Translating from Informal to Formal Language

Ambiguous Language

Negations of Multiply-Quantified Statements

Order of Quantifiers

Formal Logical Notation

Prolog

Arguments with Quantified Statements

111

(14)

Universal Modus Ponens

Use of Universal Modus Ponens in a Proof

Universal Modus Tollens

Proving Validity of Arguments with Quantified Statements

Using Diagrams to Test for Validity

Creating Additional Forms of Argument

Remark on the Converse and Inverse Errors

Elementary Number Theory and Methods of Proof

125

(74)

Direct Proof and Counterexample I: Introduction

126

(15)

Definitions

Proving Existential Statements

Disproving Universal Statements by Counterexample

Proving Universal Statements

Directions for Writing Proofs of Universal Statements

Common Mistakes

Getting Proofs Started

Showing That an Existential Statement Is False

Conjecture, Proof, and Disproof

Direct Proof and Counterexample II: Rational Numbers

141

(7)

More on Generalizing from the Generic Particular

Proving Properties of Rational Numbers

Deriving New Mathematics from Old

Direct Proof and Counterexample III: Divisibility

148

(8)

Proving Properties of Divisibility

Counterexamples and Divisibility

The Unique Factorization Theorem

Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem

156

(8)

Discussion of the Quotient-Remainder Theorem and Examples

div and mod

Alternative Representations of Integers and Applications to Number Theory

Direct Proof and Counterexample V: Floor and Ceiling

164

(7)

Definition and Basic Properties

The Floor of n/2

Indirect Argument: Contradiction and Contraposition

171

(8)

Proof by Contradiction

Argument by Contraposition

Relation between Proof by Contradiction and Proof by Contraposition

Proof as a Problem-Solving Tool

Two Classical Theorems

179

(7)

The Irrationality of √2

The Infinitude of the Set of Prime Numbers

When to Use Indirect Proof

Open Questions in Number Theory

Application: Algorithms

186

(13)

An Algorithmic Language

A Notation for Algorithms

Trace Tables

The Division Algorithm

The Euclidean Algorithm

Sequences and Mathematical Induction

199

(56)

Sequences

199

(16)

Explicit Formulas for Sequences

Summation Notation

Product Notation

Factorial Notation

Properties of Summations and Products

Change of Variable

Sequences in Computer Programming

Application: Algorithm to Convert from Base 10 to Base 2 Using Repeated Division by 2

Mathematical Induction I

215

(12)

Principle of Mathematical Induction

Sum of the First n Integers

Sum of a Geometric Sequence

Mathematical Induction II

227

(8)

Comparison of Mathematical Induction and Inductive Reasoning

Proving Divisibility Properties

Proving Inequalities

Strong Mathematical Induction and the Well-Ordering Principle

235

(9)

The Principle of Strong Mathematical Induction

Binary Representation of Integers

The Well-Ordering Principle for the Integers

Application: Correctness of Algorithms

244

(11)

Assertions

Loop Invariants

Correctness of the Division Algorithm

Correctness of the Euclidean Algorithm

Set Theory

255

(42)

Basic Definitions of Set Theory

255

(14)

Subsets

Set Equality

Operations on Sets

Venn Diagrams

The Empty Set

Partitions of Sets

Power Sets

Cartesian Products

An Algorithm to Check Whether One Set Is a Subset of Another (Optional)